Séminaire Lotharingien de Combinatoire 61A (2010), Article B61Ah

BAXTER PERMUTATIONS AND PLANE BIPOLAR ORIENTATIONS

NICOLAS BONICHON, MIREILLE BOUSQUET-MÉLOU, AND ÉRIC FUSY

Abstract. We present a simple bijection between Baxter permutations of size n and plane bipolar orientations with n edges. This bijection translates several classical parameters of permutations (number of ascents, right-to-left maxima, left-to-right minima …) into natural parameters of plane bipolar orientations (number of vertices, degree of the sink, degree of the source …), and has remarkable symmetry properties. By specializing it to Baxter permutations avoiding the pattern 2413, we obtain a bijection with non-separable planar maps. A further specialization yields a bijection between permutations avoiding 2413 and 3142 and series-parallel maps.

1. Introduction

In 1964, Glen Baxter, in an analysis context, introduced a class of permutations that now bear his name [2]. A permutation π of the symmetric group 𝔖_{n} is Baxter if one cannot find i < j < k such that π(j + 1) < π(i) < π(k) < π(j) or π(j) < π(k) < π(i) < π(j + 1). These permutations were first enumerated around 1980 [9, 28, 33, 11]. More recently, they have been studied in a slightly different perspective, in the general and very active framework of pattern avoiding permutations [7, 15, 22, 25]. In particular, the number of Baxter permutations of 𝔖_{n} having m ascents, i left-to-right maxima and j right-to-left maxima is known to be [28]:

\frac{ij}{n(n + 1)}\binom{n + 1}{m + 1} [\binom{n − i − 1}{n − m − 2}\binom{n − j − 1}{m − 1} − \binom{n − i − 1}{n − m − 1}\binom{n − j − 1}{m}] .

(1)

A few years ago, another Baxter, the physicist Rodney Baxter, studied the sum of the Tutte polynomials T_{M}(x, y) of non-separable planar maps M having a fixed size [3]. He proved that the coefficient of x^{1}y^{0} in T_{M}(x, y), summed over all rooted non-separable planar maps M having n + 1 edges, m + 2 vertices, root-face of degree i + 1 and a root-vertex of degree j + 1, was given by (1). He was unaware that these numbers had been met before (and bear “his” name), and that the coefficient of x^{1}y^{0} in T_{M}(x, y) is the number of bipolar orientations of M [24, 20]. This number is also, up to a sign, the derivative of the chromatic polynomial of M, evaluated at 1 [26].

2000 Mathematics Subject Classification. Primary 05A15; Secondary 05C30.

Key words and phrases. Baxter permutations, bipolar orientations.

The three authors are supported by the project GeoComp of the ACI Masses de Données and by the French “Agence Nationale de la Recherche”, project SADA ANR-05-BLAN-0372.

This is an amusing coincidence — are all Baxters doomed to invent independently objects that are counted by the same numbers? — which was first noticed in [7]. It is the aim of this paper to explain it via a direct bijection between Baxter permutations and plane bipolar orientations. Our bijection is non-recursive, simple to implement, and has a lot of structure: it translates many natural statistics of permutations into natural statistics of maps, and behaves well with respect to symmetries. When restricted to Baxter permutations avoiding the pattern 2413, it specializes to a bijection between these permutations and rooted non-separable planar maps, which is related to a recursive description of these maps by Dulucq et al. [14]. The key of the proofs, and (probably) the reason why this bijection has so much structure, is the existence of two isomorphic generating trees for Baxter permutations and plane orientations.

It is not the first time that intriguing connections arise between pattern avoiding permutations and planar maps^{1}. For instance, several families of permutations, including two-stack sortable permutations, are equinumerous with non-separable maps [34, 36, 14, 10]. Connected 1342-avoiding permutations are equinumerous with bicubic planar maps [6], and 54321-avoiding involutions are equinumerous with tree-rooted maps [29, 23, 4, 7]. Finally, a bijection that sends plane bipolar orientations to 3-tuples of non-intersecting paths appears in [19]. These configurations of paths are known to be in one-to-one correspondence with Baxter permutations [15]. More bijections between these three families of objects are presented in [17]. However, our construction is the first direct bijection between Baxter permutations and plane bipolar orientations. Moreover, it has interesting properties (symmetries, specializations …) that the bijections one may obtain by composing the bijections of [17] do not have.

Let us finish this introduction with the outline of the paper. After some preliminaries in Section 2, we present our main results in Section 3: we describe the bijection Φ, its inverse Φ^{−1}, and state some of its properties. In particular, we explain how it transforms statistics on permutations into statistics on orientations (Theorem 2), and how it behaves with respect to symmetry (Proposition 4). In Section 4 we introduce two generating trees, which respectively describe a recursive construction of Baxter permutations and plane bipolar orientations. We observe that these trees are isomorphic, which means that Baxter permutations and orientations have a closely related recursive structure. In particular, this isomorphism implies the existence of a (recursively defined) canonical bijection Λ between these two classes of objects, the properties of which reflect the properties of the trees (Theorem 12). We prove in Section 5 that our construction Φ coincides with the canonical bijection Λ, and then (Section 6) that our description of Φ^{−1} is correct. Section 7 is devoted to the study of two interesting specializations of Φ, and the final section presents possible further developments of this work.

1Nor is it the first time that connections between Baxter permutations and planar objects appear: several authors have discussed bijections between these permutations and certain floorplans, which have similarities with the present paper. See [1] and references therein.

−1 rev −1 rev rev  −1  rev −1

Figure 1. The symmetries of the square act on Baxter permutations.

2. Preliminaries

2.1. Baxter permutations

A permutation π = π(1) ⋯ π(n) will be represented by its diagram, that is, the set of points (i, π(i)). Hence the group of symmetries of the square, of order 8, acts on permutations of 𝔖_{n}. This group is generated (for instance) by the following two involutions:

– the inversion π ↦ π^{−1}, which amounts to a reflection in the first diagonal,

– the reversion of permutations, which maps π = π(1) ⋯ π(n) to rev(π) = π(n) ⋯ π(1).

It is clear from the definition given at the beginning of the introduction that the reversion leaves the set B_{n} of Baxter permutations of size n invariant. It is also true that the inversion leaves B_{n} invariant, so that B_{n} is invariant under the 8 symmetries of the square (Fig. 1). The invariance of B_{n} by inversion follows from an alternative description of Baxter permutations in terms of barred patterns [22], which we now describe. Given two permutations π and τ = τ_{1} ⋯ τ_{k}, an occurrence of the pattern τ in π is a subsequence π(i_{1}), …, π(i_{k}), with i_{1} < ⋯ < i_{k}, which is order-isomorphic to τ. If no such occurrence exists, then π avoids τ. Now, π avoids the barred pattern τ = τ_{1} ⋯ τ_{i−1}\overline{τ_{i}}τ_{i+1} ⋯ τ_{k} if every occurrence of τ_{1} ⋯ τ_{i−1}τ_{i+1} ⋯ τ_{k} in π is a sub-sequence of an occurrence of τ_{1} ⋯ τ_{i−1}τ_{i}τ_{i+1} ⋯ τ_{k}. Then Baxter permutations are exactly the 25¯314- and 41¯352-avoiding permutations.

We equip ℝ^{2} with the natural product order: v ≤ w if x(v) ≤ x(w) and y(v) ≤ y(w), that is, if w lies to the North-East of v. Recall that for two elements v and w of poset P, w covers v if v < w and there is no u such that v < u < w. The Hasse diagram of P is the digraph having vertex set P and edges corresponding to the covering relation. We orient these edges from the smaller to the larger element.

Given a permutation π, a left-to-right maximum (or: lr-maximum) is a value π(i) such that π(i) > π(j) for all j < i. One defines similarly rl-maxima, lr-minima and rl-minima.

2.2. Plane bipolar orientations

A planar map is a connected graph embedded in the plane with no edgecrossings, considered up to continuous deformation. A map has vertices, edges, and faces, which are the connected components of ℝ^{2} remaining after deleting the edges. The outer face is unbounded, the inner faces are bounded. The map is separable if there exists a vertex whose deletion disconnects the map. A plane bipolar orientation O is an acyclic orientation of a planar map M with a unique source s (vertex with no ingoing edge) and a unique sink t (vertex with no outgoing edge), both on the outer face (Fig. 2). The poles of O are s and t. One of the oriented paths going from s to t has the outer face on its right: this path is the right border of O, and its length is the right outer degree of O. The left outer degree is defined similarly.

O^{∗}  right vertex f source right face v O s  t

Figure 2. Left: A plane bipolar orientation, having right (left) outer degree 2 (3), and its dual. Right: Local properties of plane bipolar orientations.

It can be seen that around each non-polar vertex v of a plane bipolar orientation, the edges are sorted into two blocks, one block of ingoing edges and one block of outgoing edges: around v, one finds a sequence of outgoing edges, then a sequence of ingoing edges, and that’s it (Fig. 2, right). The face that is incident to the last outgoing edge and the first ingoing edge (taken in clockwise order) is called the right face of v. Analogously, the face that is incident to the last ingoing edge and the first outgoing edge (still in clockwise order) is called the left face of v. Dually, the border of every finite face f contains exactly two maximal oriented paths (Fig. 2, right), forming a small bipolar orientation. Its source (resp. sink) is called the source (resp. sink) of f. The other vertices of the face are respectively called right and left vertices of f: if v is a right vertex of f, then f is the left face of v.

A map M is rooted if one of the edges adjacent to the outer face is oriented, in such a way the outer face lies on its right. In this case, a bipolar orientation of M is required to have source s and sink t, where s and t are the endpoints of the root edge. As recalled above, the number of such orientations of M is the coefficient of x^{1}y^{0} in the Tutte polynomial T_{M}(x, y) (this is actually true of any graph G with an oriented edge). This number is non-zero if and only if M is non-separable [27].

A map M is bipolar if two of its vertices, lying on the outer face and respectively called the source (s) and the sink (t), are distinguished. In this case, a bipolar orientation of M is required to have source s and sink t. There is of course a simple one-to-one correspondence between rooted maps and bipolar maps, obtained by deleting the root edge and taking its starting point (resp. ending point) as the source (resp. sink). It will be convenient (and, we hope, intuitive) to use the following notation: if M is rooted, M̌ is the bipolar map obtained by deleting the root edge. If M is bipolar, M̂ is the rooted map obtained by adding a root-edge. In this case, M admits a bipolar orientation if and only if M̂ is non-separable.

Two natural transformations act on the set O_{n} of (unrooted) plane bipolar orientations having n edges. For a plane bipolar orientation O, we define mir(O) to be the mirror image of O, that is, the orientation obtained by flipping O around any line (Fig. 3). Clearly, mir is an involution.

The other transformation is duality. The dual plane orientation O^{∗} of O is constructed as shown in Fig. 2. There is a vertex of O^{∗} in each bounded face of O, and two vertices of O^{∗} (its poles) in the outer face of O. The edges of O^{∗} connect the vertices corresponding to faces of O that are adjacent to a common edge, and are oriented using the convention shown in the figure (that is, from the face located to the right of the edge to the face located to the left of the edge). Note that (O^{∗})^{∗} is obtained by changing all edge directions in O, so that duality is a transformation of order 4. The transformations mir and O ↦ O^{∗} generate a group of order 8 (Fig. 3). We shall see that our bijection Φ allows us to superimpose Figs. 1 and 3. That is, if Φ(π) = O, then Φ(π^{−1}) is mir(O) and Φ(rev(π)) is mir(O^{∗}).

mir  mir mir mir

Figure 3. A group of order 8 acts on plane bipolar orientations. The dashed edges join an orientation to its dual, the others join mirror images.

3. Main results

3.1. From Baxter permutations to plane bipolar orientations

Let π be a Baxter permutation. We first construct an embedded digraph ϕ(π) with straight edges having black and white vertices. The black vertices are the points b_{i} = (i, π(i)). The white vertices correspond to the ascents of π. More precisely, for each ascent a (i.e., π(a) < π(a + 1)), let ℓ_{a} = max{π(i) : i < a + 1 and π(i) < π(a + 1)}. The Baxter property implies that for all i > a such that π(i) > π(a), one has actually π(i) > ℓ_{a} (see Fig. 4). Create a white vertex w_{a} = (a + 1/2, ℓ_{a} + 1/2). Finally, add two more white vertices w_{0} = (1/2, 1/2) and w_{n} = (n + 1/2, n + 1/2). (In other words, we consider that π(0) = 0 and π(n + 1) = n + 1.) We define the embedded digraph ϕ(π) to be the Hasse diagram of this collection of black and white vertices for the product order on ℝ^{2}, drawn with straight edges (Fig. 5, centre). Of course, all edges point to the North-East. We will prove in Section 5 the following proposition.

Proposition 1. For all Baxter permutation π, the embedded graph ϕ(π) is planar (no edges cross), bicolored (every edge joins a black vertex and a white one), and every black vertex has indegree and outdegree 1.

These properties can be observed in the example of Fig. 5. Erasing all black vertices yields a plane bipolar orientation with source w_{0} and sink w_{n}, which we define to be Φ(π) (see Fig. 5, right). Observe that every point of the permutation gives rise to an edge of Φ(π): we will say that the point corresponds to the edge, and vice-versa. We draw the attention of the reader to the fact that the slightly vague word “correspond” has now a very precise meaning in this paper.

ℓ_{a}+1 ∅  w_{a} π(a) π(a + 1) ℓ_{a}  a a+1 ∅

Figure 4. The insertion of white vertices in ascents.

Figure 5. The Baxter permutation π = 5 3 4 9 7 8 10 6 1 2, the Hasse diagram ϕ(π) and the plane bipolar orientation Φ(π).

Theorem 2. The map Φ is a bijection between Baxter permutations and plane bipolar orientations, which transforms standard parameters as follows:

size ↔ # edges,  # ascents ↔ # non-polar vertices,
# lr-maxima ↔ left outer degree,  # rl-minima ↔ right outer degree,
# rl-maxima ↔ degree of the sink,  # lr-minima ↔ degree of the source.

By Euler’s formula, the number of inner faces of Φ(π) is the number of descents of π. Theorem 2 is proved in Section 5. It explains why the numbers (1) appear both in the enumeration of Baxter permutations [28] and in the enumeration of bipolar orientations [3].

3.2. The inverse bijection

Let O be a plane bipolar orientation, with vertices colored white. Let O′ be the bicolored oriented map obtained by inserting a black vertex in the middle of each edge of O (Fig. 6, left). Recall how the in- and out-going edges are organized around the vertices of O (Fig. 2, right). Let T_{x} be obtained from O′ by retaining, at each white vertex, only the first incoming edge in clockwise order (at the sink, we only retain the incoming edge lying on the right border). All vertices of T_{x}, except from the source of O, have now indegree one, and can be reached from the source. Hence T_{x} is a plane tree rooted at the source.

Similarly, let T_{y} be obtained from O′ by retaining, at each white vertex, the last incoming edge in clockwise order (at the sink, we only retain the incoming edge lying on the left border). Then T_{y} is also a spanning tree of O′, rooted at the source of O.

Label the black vertices of T_{x} in prefix order, walking around T_{x} clockwise. Label the black vertices of T_{y} in prefix order, but walking around T_{y} counterclockwise. For every black vertex v of O′, create a point at coordinates (x(v), y(v)), where x(v) and y(v) are the labels of v in T_{x} and T_{y} respectively. Let Ψ(O) be the collection of points thus obtained (one per edge of O).

Theorem 3. For every plane bipolar orientation O, the set of points Ψ(O) is the diagram of the Baxter permutation Φ^{−1}(O).

This can be checked in the examples of Figs. 5–6, and is proved in Section 6. The construction Ψ is closely related to an algorithm introduced in [13] to draw planar orientations on a grid.

3.3. Symmetries and specializations

3.3.1. Symmetries. We now describe how the symmetries of the square, which act in a natural way on Baxter permutations, are transformed through our bijection.

Proposition 4. Let π be a Baxter permutation and O = Φ(π). Then

Φ(π^{−1}) = mir(O) and Φ(rev(π)) = mir(O^{∗}).

Figure 6. A bipolar orientation O, and the trees T_{x} and T_{y} used to compute the coordinates of the points of Ψ(O).

\begin{matrix}π^{−1} \\ ρ(π) \end{matrix}

Figure 7. The Hasse diagram and the bipolar orientations associated to π^{−1} and ρ(π), where π = 5 3 4 9 7 8 10 6 1 2 is the Baxter permutation of Fig. 5.

By combining both properties, this implies

O^{∗} = Φ(ρ(π)),

where ρ is the clockwise rotation by 90 degrees.

Moreover, if the point p = (i, π(i)) of π corresponds (via the bijection Φ) to the edge e of O, then the point (π(i), i) of π^{−1} corresponds to the edge mir(e) in mir(O), and the point ρ(p) in ρ(π) corresponds to the dual edge of e in O^{∗}.

The symmetry properties dealing with the inverse and the rotation are illustrated in Fig. 7. The first one is easily proved from the definition of ϕ and Φ. Indeed, it is clear from Fig. 4 that π has an ascent at a if and only if π^{−1} has a ascent at ℓ_{a}, and that the white dots of the embedded graph ϕ(π^{−1}) are the mirror images of the white dots of ϕ(π) across the first diagonal. Of course this holds for black vertices as well. Hence the Hasse diagram ϕ(π^{−1}) is obtained by flipping the diagram ϕ(π) about the first diagonal. The first property follows, as well as the correspondence between the point (π(i), i) and the edge mir(e).

The second property is non-trivial, and will be proved in Section 5.

3.3.2. Specializations. As recalled in Section 2, a bipolar map M has a bipolar orientation if and only if the rooted map M̂ is non-separable.

Take a Baxter permutation π and the bipolar orientation Φ(π), with poles s and t. Let M be the underlying bipolar map, and define Φ̂(π) to be the rooted non-separable M̂. We still call s and t the source and the sink of the rooted map. It is not hard to see that different permutations may give the same map. However, the following is true.

Proposition 5. The restriction of Φ̂ to Baxter permutations avoiding the pattern 2413 (that is, to permutations avoiding 2413 and 41¯352) is a bijection between these permutations and rooted non-separable planar maps, which transforms standard parameters as follows:

– if π has length n, m ascents, i lr-maxima, j rl-maxima, k lr-minima and ℓ rl-minima,

– then Φ̂(π) has n + 1 edges, m non-polar vertices, a sink of degree j + 1, a source of degree k + 1 and the face that lies to the right (resp. left) of the root edge has degree i + 1 (resp. ℓ + 1).

This proposition is proved in Section 7.1. The fact that permutations avoiding 2413 and 41¯352 are equinumerous with non-separable planar maps was already proved in [14], by exhibiting isomorphic generating trees for these two classes. This isomorphism could be used to describe a recursive bijection between permutations and maps. It turns out that, up to simple symmetries, our direct, non-recursive bijection, is equivalent to the one that is implicit in [14]. This is explained in Section 7.3.

Observe that, if π is a Baxter permutation avoiding 2413, then π^{−1} is a Baxter permutation avoiding 3142. As Φ(π^{−1}) = mir(Φ(π)), the restriction of Φ̂ to Baxter permutations avoiding 3142 is also a bijection with non-separable planar maps. We now describe what happens when we restrict Φ̂ to permutations avoiding both 2413 and 3142 (such permutations are always Baxter as they obviously avoid the barred patterns 25¯314 and 41¯352).

Proposition 6. The restriction of Φ̂ to permutations avoiding the patterns 2413 and 3142 is a bijection between these permutations and rooted series-parallel maps, which transforms the standard parameters in the same way as in Proposition 5.

We say that a rooted non-separable map M is series-parallel if it does not contain the complete graph K_{4} as a minor. The terminology is slightly misleading: the map that can be constructed recursively using the classical series and parallel constructions (Fig. 17) is not M itself, but the bipolar map M̌. This is the case for the bipolar map of Fig. 5, and one can check that the associated permutation avoids both 2413 and 3142. Details are given in Section 7.2, with the proof of Proposition 6.

It is easy to count series-parallel maps, using their recursive description (we are in the simple framework of decomposable structures [18]). The series that counts them by edges is (up to a factor t accounting for the root edge) the generating function of large Schröder numbers [31, Exercise 6.39],

S(t) = t \frac{1 − t − \sqrt{1 − 6t + t^{2}}}{2} = ∑_{n≥1} t^{n+1} ∑_{k=0}^{n} \frac{(n + k)!}{k!(k + 1)!(n − k)!}.

Thus the number of permutations of size n avoiding 2413 and 3142 is the nth Schröder number, as was first proved in [35]. See also [22], where this result is proved via a bijection with certain trees.

4. Generating trees

A generating tree is a rooted plane tree with labelled nodes satisfying the following property: if two nodes have the same label, the lists of labels of their children are the same. In other words, the (ordered) list of labels of the children is completely determined by the label of the parent. The rule that tells how to label the children of a node, given its label, is called the rewriting rule of the tree. In particular, the tree T with root (1, 1) and rewriting rule:

(i, j) ⇝ \{ \begin{matrix}(1, j + 1), & (2, j + 1), & … & (i, j + 1), \\ (i + 1, j), & … & (i + 1, 2), & (i + 1, 1), \end{matrix}  (2)

will be central in the proofs of our results. The first three levels of this tree are shown in Fig. 8, left. In this section, we describe a generating tree for Baxter permutations, and another one for plane orientations, which are both isomorphic to T . The properties of these trees, combined with this isomorphism, imply the existence of a canonical bijection between Baxter permutations and plane orientations satisfying the conditions stated in Theorem 2. In the next section, we will prove that this bijection coincides with our map Φ.

4.1. A generating tree for Baxter permutations

Take a Baxter permutation π of length n + 1, and remove the value n + 1: this gives another Baxter permutation, denoted \overline{π}, of length n. Conversely, given σ ∈ B_{n}, it is well known, and easy to see, that the permutations π such that \overline{π} = σ are obtained by inserting the value n + 1:

• either just before an lr-maximum of σ,

• or just after an rl-maximum of σ. This observation was used already in the first paper where Baxter permutations were counted [9]. We write π = L_{k}(σ) if π is obtain by inserting n + 1 just before the kth lr-maximum of σ, and π = R_{k}(σ) if π is obtain by inserting n + 1 just after the kth rl-maximum of σ (with the convention that the first lr-maximum is σ(n)). We henceforth distinguish left and right insertions, or L- and R-insertions for short. This is refined by talking about (L, k)-insertions (and (R, k)-insertions) when we need to specify the position of the insertion.

This construction allows us to display Baxter permutations as the nodes of a generating tree T_{b}. The root is the unique permutation of size 1, and the children of a node σ having i lr-maxima and j rl-maxima are, from left to right,

L_{1}(σ), L_{2}(σ), …, L_{i}(σ), R_{j}(σ), …, R_{2}(σ), R_{1}(σ).

Hence we find at level n the permutations of B_{n}. The first layers of T_{b} are shown on the right of Fig. 8. Observe that, if σ has i lr-maxima and j rl-maxima, then L_{k}(σ) has k lr-maxima and j + 1 rl-maxima, while R_{k}(σ) has i + 1 lr-maxima and k rl-maxima. In other words, if we replace in the tree T_{b} every permutation by the pair (i, j) giving the number of lr-maxima and rl-maxima, we obtain the generating tree T defined by (2). That is, the following holds true.

Proposition 7. The generating tree T_{b} of Baxter permutations is isomorphic to the tree T defined by (2).

(1, 3) (2, 2) (2, 1)  .3.2.1..2.3.1. .2 1.3.  .1.3.2..1.2.3. .3.1 2. .1. (1, 1)(1, 2) (2, 2) (3, 1)  .1.2. .2.1. (2, 1) (1, 2)

Figure 8. The generating tree T with rewriting rule (2), and the generating tree T_{b} of Baxter permutations. The dots represent the possible insertion positions for the new maximal element.

We now observe a property of the tree T_{b} that will be crucial to prove one of the symmetry properties of our bijection.

Proposition 8. The tree obtained from T_{b} by replacing each permutation π by rev(π) coincides with the tree obtained by reflecting T_{b} in a (vertical) mirror.

In other words, if the sequence of insertions (S_{1}, k_{1}), (S_{2}, k_{2}), …, (S_{n−1}, k_{n−1}), where S_{i} ∈ {L, R} and k_{i} ∈ ℕ, leads from the permutation 1 to the permutation π ∈ B_{n}, then the sequence (S̄_{1}, k_{1}), (S̄_{2}, k_{1}), …, (S̄_{n−1}, k_{n−1}) obtained by replacing all L’s by R’s and vice-versa leads from 1 to rev(π). Proof of Proposition 8. It suffices to observe that

rev(L_{k}(π)) = R_{k}(rev(π)) and rev(R_{k}(π)) = L_{k}(rev(π)).

4.2. A generating tree for plane bipolar orientations

Let O be a plane bipolar orientation with n + 1 edges. Let e be the last edge of the left border of O, and let v be its starting point. The endpoint of e is the sink t. Perform the following transformation: if v has outdegree 1, contract e, otherwise delete e. This gives a new plane bipolar orientation, denoted \overline{O}, having n edges. Indeed, the degree condition on v guarantees that the contraction never creates a cycle and that the deletion never creates a sink.

Conversely, let P be an orientation with n edges. We want to describe the orientations O such that \overline{O} = P. Our discussion is illustrated in Fig. 9. Let i be the left outer degree of P, and j the (in)degree of the sink t of P. Let v_{1} = s, v_{2}, …, v_{i}, v_{i+1} = t be the vertices of the left border, visited from s to t. Denote e_{1}, e_{2}, …, e_{j} the edges incident to t, from right to left (the infinite face is to the right of e_{1} and to the left of e_{j}). The orientations O such that \overline{O} = P are obtained by adding an edge e whose contraction or deletion gives P. This results in two types of edge-insertion:

• Type L. For k ∈ ⟦1, i⟧ = {1, 2, …, i}, the orientation L_{k}(P) is obtained by adding an edge from v := v_{k} to t, having the infinite face on its left.

• Type R. For k ∈ ⟦1, j⟧, the orientation R_{k}(P) is constructed as follows. Split the vertex t into two neighbour vertices t and v, and re-distribute the edges adjacent to t: the edges e_{1}, e_{2}, …, e_{k−1} remain connected to t, while e_{k}, e_{k+1}, …, e_{j} are connected to v. Add an edge from v to t.

In both cases, the last edge of the left border of the resulting orientation O joins v to t. After an L-insertion, v has outdegree 2 or more, so that e will be deleted in the construction of \overline{O}, giving the orientation P. After an R-insertion, v has outdegree 1, so that e will be contracted in the construction of \overline{O}, giving the orientation P. The fact that we use, as in the construction of Baxter permutations, the notation L_{k} and R_{k}, is of course not an accident.

v_{2}  t s = v_{1}  t v_{3} v = v_{2} v_{1} Type L (k = 2) Type R (k = 3) e_{4}e_{3}  t v v_{3}  e_{2} e_{1}  e_{1} e_{3} e_{4} e_{2}

Figure 9. Inserting a new edge in a plane bipolar orientation (i = 3, j = 4).

We can now define the generating tree T_{o} of plane bipolar orientations: the root is the unique orientation with one edge, and the children of a node P having left outer degree i and sink-degree j are, from left to right,

L_{1}(P), L_{2}(P), …, L_{i}(P), R_{j}(P), …, R_{2}(P), R_{1}(P).

Hence we find at level n the orientations with n edges. The first levels of this tree are shown in Fig. 10. Observe that, if P has left outer degree i and sink-degree j, then L_{k}(P) has left outer degree k and sink-degree i + 1, while R_{k}(P) has left outer degree i + 1 and sink-degree k. In other words, if we replace in the tree T_{o} every orientation by the pair (i, j) giving the left outer degree and the sink-degree, we obtain the generating tree T defined by (2).

Proposition 9. The generating tree T_{o} of plane bipolar orientations is isomorphic to the tree T defined by (2).

Figure 10. The generating tree T_{o} of plane bipolar orientations.

We finally observe a property of the tree T_{o} that is the counterpart of Proposition 8. Recall the definitions of the dual and mirror orientations, given in Section 2.2.

Proposition 10. The tree obtained from T_{o} by replacing each orientation O by mir(O^{∗}) coincides with the tree obtained by reflecting T_{o} in a (vertical) mirror.

In other words, if the sequence of insertions (S_{1}, k_{1}), (S_{2}, k_{2}), …, (S_{n−1}, k_{n−1}), where S_{i} ∈ {L, R} and k_{i} ∈ ℕ, leads from the root of T_{o} to the orientation O ∈ O_{n}, then the sequence (S̄_{1}, k_{1}), (S̄_{2}, k_{1}), …, (S̄_{n−1}, k_{n−1}) obtained by replacing all L’s by R’s and vice-versa leads from the root to mir(O^{∗}). Proof of Proposition 10. It suffices to observe that

mir((L_{k}(O))^{∗}) = R_{k}(mir(O^{∗})) and mir((R_{k}(O))^{∗}) = L_{k}(mir(O^{∗})).

This should be clear from Fig. 11, which shows that applying L_{k} to O boils down to applying R_{k} to mir(O^{∗}), and that conversely, applying R_{k} to O boils down to applying L_{k} to mir(O^{∗}). Note that this figure only shows O^{∗}, and not its mirror image mir(O^{∗}).

We conclude with an observation that will be useful in the proof of Proposition 4.

Remark 11. Take a plane bipolar orientation O having n edges, and label these edges with 1, 2, …, n in the order where they were created in the generating tree. Fig. 11 shows that when the edge e is added to O in the generating tree, the edge mir(e^{∗}) is added to mir(O^{∗}) (we denote by e^{∗} the dual edge of e). Consequently, for all edges e of O, the label of e in O coincides with the label of mir(e^{∗}) in mir(O^{∗}).

R_{3} O^{∗}  O  L_{2}

Figure 11. How the orientation O^{∗} is changed when L_{k} or R_{k} is applied to O.

4.3. The canonical bijection

We have seen that the generating trees T_{b} and T_{o} are both isomorphic to the tree (2) labelled by pairs (i, j). This gives immediately a canonical bijection Λ between Baxter permutations and plane bipolar orientations: this bijection maps the permutation 1 to the one-edge orientation, and is then defined recursively by

Λ(L_{k}(π)) = L_{k}(Λ(π)) and Λ(R_{k}(π)) = R_{k}(Λ(π)).

In other words, if π is obtained in the Baxter tree by the sequence of insertions (S_{1}, k_{1}), (S_{2}, k_{2}), …, (S_{n−1}, k_{n−1}), then Λ(π) is the orientation obtained by the same sequence of insertions in the tree of orientations.

Theorem 12. The map Λ is a bijection between Baxter permutations and plane bipolar orientations, which transforms standard parameters as follows:

size ↔ # edges,  # ascents ↔ # non-polar vertices,
# lr-maxima ↔ left outer degree,  # rl-minima ↔ right outer degree,
# rl-maxima ↔ degree of the sink,  # lr-minima ↔ degree of the source.

Moreover, if π is a Baxter permutation and O = Λ(π), then

Λ(rev(π)) = mir(0^{∗}).

Proof. The properties of Λ dealing with the size, lr-maxima and rl-maxima follow directly from the isomorphism of the trees T_{b}, T_{o} and T . The next three properties are proved by observing that the relevant parameters evolve in the same way in the recursive construction of Baxter permutations and plane bipolar orientations. Indeed, if we replace every node π of T_{b} by (i, j; m, k, ℓ), where i, j, m, k, ℓ are respectively the number of lr-maxima, rl-maxima, ascents, lr-minima and rl-minima of π, we obtain the generating tree with root (1, 1; 0; 1, 1) and rewriting rule:

(i, j; m, k, ℓ) ⇝ \{ \begin{matrix}(1, j + 1; m, k + 1, ℓ), (2, j + 1; m, k, ℓ), …, (i, j + 1; m, k, ℓ), \\ (i + 1, j; m + 1, k, ℓ), …, (i + 1, 2; m + 1, k, ℓ), \\ (i + 1, 1; m + 1, k, ℓ + 1). \end{matrix}

(3)

Phrased in words, the number of ascents increases by 1 in an R-insertion, and is unchanged otherwise. The number of lr-minima is only changed if we perform an (L, 1)-insertion (and then it increases by 1), and the number of rl-minima is only changed if we perform an (R, 1)-insertion (and then it increases by 1).

It is not hard to see that one obtains the same tree by replacing every orientation O of T_{o} by (i, j; m, k, ℓ), where i, j, m, k, ℓ are respectively the left outer degree, the sink-degree, the number of non-polar vertices, the source-degree and the right outer degree. That is, the number of vertices only increases (by 1) in an Rinsertion, the degree of the source only increases (by 1) in an (L, 1)-insertion and the right outer degree only increases (by 1) in an (R, 1)-insertion.

Let us finally prove the symmetry property. Let (S_{1}, k_{1}), (S_{2}, k_{2}), …, (S_{n−1}, k_{n−1}) be the sequence of insertions that leads to π in the Baxter tree. By definition of Λ, this sequence leads to O = Λ(π) in the tree of orientations. By Propositions 8 and 10, the sequence (S̄_{1}, k_{1}), (S̄_{2}, k_{1}), …, (S̄_{n−1}, k_{n−1}) obtained by swapping the L’s and the R’s leads respectively to rev(π) and mir(O^{∗}) in the trees T_{b} and T_{o}. By definition of Λ, this means that mir(O^{∗}) = Λ(rev(π)).

4.4. A lattice path interpretation of the generating tree

Baxter permutations have been shown to be in bijection with certain 3-tuples of non-intersecting lattice paths [33, 16]. We describe here a simple generating tree for these paths, which is again isomorphic to the tree T defined by (2).

Let n ≥ 0, and consider 3-tuples of non-intersecting paths consisting of north and east steps, starting from the points (−1, 1), (0, 0) and (1, −1) and ending at 3 consecutive points on the diagonal x + y = n, say (m − 1, n − m + 1), (m, n − m) and (m+1, n−m−1). These path configurations can be displayed as the nodes of a generating tree as follows. The root of the tree is labelled by the configuration formed of 3 paths of length 0. If C is a configuration of paths of length n > 0, the parent of C in the tree is obtained by contracting the last north (resp. east) step in every path if the central path ends with a north (resp. east) step. The first levels of the tree are shown of Fig. 12.

Conversely, let C be a configuration of length n − 1. Assume the top path ends with (exactly) i − 1 east steps, and the bottom path with (exactly) j − 1 north steps, with i, j ≥ 1. For 1 ≤ k ≤ i, let L_{k}(C) be obtained by inserting 3 north steps in C as follows:

• one north step is inserted in the last horizontal run of the top path, at distance k − 1 from the endpoint,

• one north step is added at the end of the central path, and one at the end of the bottom path.

Figure 12. The generating tree of path configurations.

One defines analogously R_{k}(C), for 1 ≤ k ≤ j, by inserting 3 east steps in the configuration (at the end of the top and central paths, and in the last vertical run of the bottom path). Then the L_{k}(C) and R_{k}(C) are the children of the node C in the tree. We order them from left to right as follows: L_{1}(C), …, L_{i}(C), R_{j}(C), …, R_{1}(C). It is not hard to see that, if we replace every configuration C by the associated pair (i, j), we obtain once again the tree T .

This construction induces a canonical bijection between Baxter permutations of size n and configurations of length n − 1. We have not checked whether it is related to one of the already published ones [33, 16]. Of course, the number of lr-maxima (resp. rl-maxima) of the permutation becomes the parameter i (resp.

j) in the path configuration. Moreover, the number of ascents of the permutation is the number m of east steps in each path of the configuration. Indeed, it is easy to check that this number evolves as described in (3). (The remaining parameters, k and ℓ, count certain “bottlenecks” in the path configuration.)

From these observations, it follows that the formula (1) giving the number of Baxter permutations of length n, having m ascents, i lr-maxima and j rl-maxima results now from a simple application of the Gessel–Viennot determinantal formula for configurations of non-intersecting paths [21]. This was already used in [16].

5. The mapping Φ is the canonical bijection

In the previous section, we have described recursively a bijection Λ that implements the isomorphism between the generating trees T_{b} and T_{o}, and we have shown that it has some interesting properties (Theorem 12). We now prove that the mapping Φ defined in Section 3 coincides with this canonical bijection Λ. Simultaneously, we prove the properties of the map ϕ stated in Proposition 1.

Proposition 13. For each Baxter permutation π, the embedded graph ϕ(π) is planar and bicolored, with black vertices of indegree and outdegree 1. Moreover, Φ(π) = Λ(π).

Proof. The proof is by induction on the size of π. Both statements are obvious for π = 1. Now assume that the proposition holds for Baxter permutations of size n. Let π ∈ B_{n+1}, and let σ be the parent of π in the tree T_{b}. Then either π = L_{k}(σ), or π = R_{k}(σ) for some k. Let O = Φ(σ) = Λ(σ). We want to prove that ϕ(π) satisfies the required conditions and that Φ(π) = L_{k}(O) (or that Φ(π) = R_{k}(O) in the case of an R-insertion). Both statements follow from a careful observation of

insertion stretch  b v_{k} a + 1 a w_{a} = v_{k} w_{0} = v_{1}  v_{i+1} w_{n} = v_{i+1}  v_{1}

Figure 13. How the graph ϕ(σ) changes during an (L, k)-insertion.

how ϕ(σ) is changed into ϕ(π) as n + 1 is inserted in σ. Some readers will think that looking for two minutes at Figs. 13 and 14 is sufficient to get convinced of the result. For the others, we describe below in greater detail what happens during the insertion of n + 1 in σ. Essentially, we describe the generating tree whose nodes are the embedded graphs ϕ(π), for π ∈ B_{n}. Case 1. The permutation π is obtain by left insertion, that is, π = L_{k}(σ). Let a + 1 be the abscissa of the kth lr-maximum of σ. Let i be the number of lrmaxima in σ. Then the left border of O has i + 1 vertices, ranging from v_{1} = w_{0} to v_{i+1} = w_{n}. The vertex w_{a} is the kth vertex v_{k} of the left border of O. As n + 1 is inserted in σ, the ascents of σ become ascents of π, and no new ascent is created. All the vertices that occur in ϕ(σ) occur in ϕ(π), but some of them are translated: the white vertex w_{n} moves from (n+1/2, n+1/2) to (n+3/2, n+3/2), and all vertices located at abscissa x ≥ a + 1 move one unit to the right. These translations, illustrated by the first two pictures of Fig. 13, stretch some edges but do not affect the covering relations among vertices. We claim that they also do not affect planarity. Indeed, it is easy to see that the Hasse diagram of a set S of points in the plane having distinct abscissas and ordinates, has no crossing if and only if S avoids the pattern 21¯354 [8]. Clearly, this property is not affected by translating to the right the rightmost points of S.

After the translation operations, one new vertex is created: a black vertex b = (a + 1, n + 1) corresponding to the new value n + 1 in π. We need to study how this affects the covering relations. That is, which vertices cover b, and which vertices are covered by b?

As b lies at ordinate n + 1, it is only covered by w_{n}. This results in a new bicolored edge from b to w_{n}, which lies sufficiently high not to affect the planarity. Hence b has outdegree 1.

It remains to see which vertices b covers. Clearly it covers w_{a}. But then all the vertices lying to the South-West of b are actually smaller than w_{a} for our ordering (because σ(a) was an lr-maximum), so that b covers no vertex other than w_{a}. This means it has indegree 1.

To summarize, one goes from ϕ(σ) to ϕ(π), where π = L_{k}(σ), by • stretching some edges by a translation of certain vertices,

• inserting one new vertex, b = (a + 1, n + 1),

• adding an edge from w_{a} to b, and another one from b to w_{n}.

The resulting graph is still bicolored, with black vertices of indegree and outdegree 1. The planarity is preserved as all the changes occur to the North-West of the lr-maxima of σ. Finally, recall that the orientation O = Φ(σ) and Φ(π) are respectively obtained by erasing the black vertices in ϕ(σ) and ϕ(π). It should now be clear that Φ(π) is exactly the result of an (L, k)-insertion in O: denoting v_{1}, …, v_{i}, v_{i+1} = w_{n} the vertices of the left border of Φ(σ), one has simply added a new edge from v_{k} = w_{a} to the sink w_{n} in the outer face of Φ(σ). Case 2. The permutation π is obtain by right insertion, that is, π = R_{k}(σ). Let m_{1}, …, m_{j} be the points of the diagram of σ corresponding to its rl-maxima, from right to left. Then the sink t = w_{n} of O has degree j, and for all ℓ, the ℓth edge that arrives at t (from right to left) corresponds to the point m_{ℓ} of σ via the correspondence Φ. As noticed in the proof of Theorem 12, the ascents of σ become ascents of π, and a new ascent occurs at position a, if the kth rl-maximum of σ is σ(a). In particular, all the vertices that occur in ϕ(σ) occur in ϕ(π). However, some of them are translated: the white vertex w_{n} moves from (n+1/2, n+1/2) to (n + 3/2, n + 3/2), while the vertices located at abscissa x ≥ a + 1 move one unit to the right. These translations, illustrated by the first two pictures of Fig. 14, do not affect the planarity nor the covering relations.

Then two new vertices are created: a black vertex b = (a+1, n+1) corresponding to the new value n+1 in π, and a white vertex w = (a+1/2, n+1/2) corresponding to the new ascent. We need to study how they affect the covering relations. That is, which vertices cover w or b, and which vertices are covered by w or b?

As w and b lie respectively at ordinate n + 1/2 and n + 1, it is easily seen that w is only covered by b, which is only covered by w_{n}. This results in two new bicolored edges, from w to b, and from b to w_{n}, which lie sufficiently high not to affect the planarity. Moreover, b does not cover any vertex other than w. We have thus proved that b has indegree and outdegree 1.

It remains to see which vertices w covers. These vertices were covered in ϕ(σ) by vertices that are now larger than w. But w_{n} is the only vertex larger than w that was already in ϕ(σ). Since by assumption, ϕ(σ) is bicolored, the vertices covered by w are black, and hence were rl-maxima in σ. The vertices m_{1}, …, m_{k−1} are still covered by w_{n} (they lie to the right of w), but m_{k}, …, m_{j} are covered by w (Fig. 14, right).

To summarize, one goes from ϕ(σ) to ϕ(π), where π = R_{k}(σ), by

• stretching some edges by a translation of certain vertices,

• inserting two new vertices, w = (a + 1/2, n + 1/2) and b = (a + 1, n + 1),

• adding an edge from w to b, and another one from b to w_{n},

• re-directing the edge that starts from m_{r} so that it points to w rather than w_{n}, for r ≥ k.

The resulting graph is still bicolored, with black vertices of indegree and outdegree 1. The planarity is preserved as all the changes occur to the North-East of the rl-maxima of σ. Finally, recall that the orientations O = Φ(σ) and Φ(π) are respectively obtained by erasing the black vertices in ϕ(σ) and ϕ(π). It should now

stretch m_{1} a m_{k}  insertion  b m_{k}a m_{j}  w m_{j} w_{n}  w_{n} e_{k} e_{1}

Figure 14. How the graph ϕ(σ) changes during an (R, k)-insertion.

be clear that Φ(π) is exactly the result of an (R, k)-insertion in O: the leftmost j − k + 1 edges that were pointing to w_{n} are now pointing to the new white vertex w.

We now know that Φ coincides with the canonical bijection Λ, whose properties were stated in Theorem 12. This implies Theorem 2, and the second symmetry property of Proposition 4. The first property was proved just after the statement of this proposition, as well as the correspondence between the point (π(i), i) and the edge mir(e). To conclude the proof of Proposition 4, it remains to prove that ρ(p) corresponds to the edge e^{∗}.

Observe that, as n+1 is inserted in σ to form the permutation π, the edge e that is added to O = Φ(σ) to form Φ(π) corresponds to the point of ordinate n + 1 in the diagram of π. Hence, the edge labelling introduced in Remark 11 boils down to labelling every edge of Φ(π) by the ordinate of the corresponding point of π. Consequently, Remark 11 can be reformulated as follows: if the point p = (i, π(i)) corresponds to the edge e, then the point rev(p) (which is the point of rev(π) with the same ordinate as p) corresponds to mir(e^{∗}). Combining this correspondence with the first one (which deals with π^{−1} and mir(O)) gives the final statement of Proposition 4.

6. The inverse bijection

It is now an easy task to prove that the map Ψ described in Section 3.2 is indeed the inverse of the bijection Φ. As we already know that Φ is bijective, it suffices to prove that Ψ(Φ(π)) = π for all Baxter permutation π. For a Baxter permutation π, we denote by O the orientation Φ(π) and by O′ the bicolored oriented map obtained from O by adding a black vertex in the middle of each edge. Recall that ϕ(π) is an embedding of O′. This allows us to consider the trees T_{x} and T_{y} as embedded in ℝ^{2}. In particular, the edge of T_{x} joining a white vertex v to its parent is the steepest edge ending at v in ϕ(π). See Fig. 15, left.

Every point of the diagram of π corresponds to an edge of O. Thus the black vertices that we have added to form O′ are in one-to-one correspondence with the points of π. This allows us to identify the black vertices of O′ with the points of π. We want to check that the order induced on these vertices by the

w  u v Path in ϕ(π) from v to u

Figure 15. Left: The tree T_{x} obtained from the Baxter permutation π = 5 3 4 9 7 8 10 6 1 2 Right: Why the prefix order and the abscissa order coincide.

clockwise prefix order of T_{x} coincides with the order induced by the abscissas of the points. Similarly, we want to check that the order induced on the vertices by the counterclockwise prefix order of T_{y} coincides with the order induced by the ordinates of the points. As the constructions of ϕ and Ψ are symmetric with respect to the reflection in the first diagonal, it suffices to prove the statement for the tree T_{x}. Since we are comparing two total orders, it suffices to prove that, if the vertex v comes just after the vertex u in the prefix order of T_{x}, then v lies to the right of u in ϕ(π). Two cases occur.

If u is not a leaf of T_{x}, it has a (unique, white) child, the first child of which is v. As all edges in ϕ(π) point North-East, v is to the right of u.

If u is a leaf (Fig. 15, right), let w be its closest ancestor (necessarily white) that has at least one child to the right of the branch leading to u. By definition of the prefix order, the first of these children is v. Observe that, by definition of Ψ, the path of T_{x} joining u to its ancestor w is the steepest (unoriented) path of ϕ(π) leading from u to w. More precisely, it is obtained by starting at u, and choosing at each time the steepest down edge, until w is reached. Assume v is to the left of u. Then it is also below u, and, as ϕ(π) is a Hasse diagram, there exist paths in ϕ(π) from v to u. Take the steepest of these: that is, start from u, take the steepest down edge that can be extended into a down path ending at v, and iterate until v is reached. Finally, add the edge (w, v): this gives a path joining w and u that is steeper than the one in T_{x}, a contradiction. Hence v lies to the right of u.

7. Specializations

7.1. Baxter permutations avoiding 2413 and rooted non-separable maps

The aim of this subsection is to prove Proposition 5: if we restrict Φ to Baxter permutations avoiding 2413, add a root-edge from the source to the sink, and forget the orientations of all (non-root) edges, we obtain a bijection with rooted nonseparable planar maps, which transforms standard parameters of permutations into standard parameters of maps. Our first objective will be to describe the orientations corresponding via Φ to 2413-avoiding Baxter permutations. Recall that the faces of a bipolar orientation have left and right vertices (Fig. 2). The following definition is illustrated in Fig. 16(a).

Definition 14. Given a plane bipolar orientation O, a right-oriented piece (ROP) is a 4-tuple (v_{1}, v_{2}, f_{1}, f_{2}) formed of two vertices v_{1}, v_{2} and two inner faces f_{1}, f_{2} of O such that:

• v_{1} is the source of f_{1} and is a left vertex of f_{2},

• v_{2} is the sink of f_{2} and is a right vertex of f_{1}.

A left-oriented piece (LOP) is defined similarly by swapping ’left’ and ’right’ in the definition of a ROP. Consequently, a ROP in O becomes a LOP in mir(O) and vice-versa.

i_{2} i_{1} π(i_{2}) π(i_{1})  i_{4} i_{3} π(i_{4}) π(i_{3}) f_{1} v_{1}  f_{2}  f_{1} v_{1}  f_{2} e_{2} v_{2}  v_{2}  f_{2} v_{1}  v_{2} e_{4} e_{3} e_{1}  f_{1} (b) (a)  (c)

Figure 16. (a) A right-oriented piece (ROP) in a plane bipolar orientation. (b) The four distinguished edges of a ROP. (c) A minimal pattern 2413 in a Baxter permutation yields a ROP in the associated plane bipolar orientation. The dashed edges come from ϕ(ρ(π)).

Proposition 15 ([30]). Every bipolar planar map admits a unique bipolar orientation with no ROP, and a unique plane bipolar orientation with no LOP.

The set of bipolar orientations of a fixed bipolar map can actually be equipped with the structure of a distributive lattice, the minimum (resp. maximum) of which is the unique orientation with no ROP (resp. LOP) [30].

We can now characterize the image by Φ of 2413-avoiding Baxter permutations.

Proposition 16. A Baxter permutation π contains the pattern 2413 if and only if the bipolar orientation O = Φ(π) contains a ROP. Analogously, π contains the pattern 3142 if and only if O contains a LOP.

Simple examples are provided by π = 25314 and π = 41352. The corresponding orientations (with a root edge added) are those of Fig. 18.

Proof. Assume O contains a ROP (v_{1}, v_{2}, f_{1}, f_{2}). Denote e_{1}, e_{2}, e_{3}, e_{4} the four edges shown in Fig. 16(b). To each of them corresponds a point p_{i} of the diagram of π, with 1 ≤ i ≤ 4. We will prove that these points form an occurrence of 2413.

Recall that the order of the abscissas and ordinates of these points is obtained from the trees T_{x} and T_{y} defined in Section 3.2. Denote by P (e) the path of T_{x} that joins (the middle of) the edge e ∈ O to the source. By definition of T_{x}, the point of π corresponding to e occurs in π to the left of the point corresponding to e′ if and only if either e lies on the path P (e′), or P (e) is on the left of P (e′) when the two paths meet. From this observation and the configuration of a ROP, it is clear that the x-order of the points p_{i} is p_{1}, p_{2}, p_{3}, p_{4}. By similar arguments, the y-order of these points is p_{3}, p_{1}, p_{4}, p_{2}. Hence they form an occurrence of 2413.

Conversely, let π be a Baxter permutation containing an occurrence p_{1}, p_{2}, p_{3}, p_{4} of the pattern 2413. This means that p_{j} = (i_{j}, π(i_{j})) with i_{1} < i_{2} < i_{3} < i_{4} and π(i_{3}) < π(i_{1}) < π(i_{4}) < π(i_{2}). We can assume that the p_{i} form a minimal occurrence of 2413: that is, the bounding rectangle R of the p_{i} does not contain any other occurrence of the pattern. Then it is easy to see that no point of R lies between columns i_{1} and i_{2} (this would create a smaller pattern). Similarly, no point of R lies between columns i_{3} and i_{4}, or between rows π(i_{3}) and π(i_{1}), or between rows π(i_{4}) and π(i_{2}). The empty areas are shaded in Fig. 16(c).

We now want to prove that O contains a ROP. Consider an oriented path going from p_{1} to p_{2} in the embedded Hasse diagram ϕ(π). As ϕ(π) is bipartite and there is no point in R between columns i_{1} and i_{2}, this path has length 2, and goes from p_{1} to p_{2} via a unique white vertex, which we denote by v_{1}. As the p_{i} have inand out-degree 1, there is no other path between p_{1} and p_{2}. Similarly, there is a unique oriented path going from p_{3} to p_{4}, which has length 2. Denote by v_{2} the (unique) white vertex of this path.

We will exhibit a ROP whose vertices are v_{1} and v_{2}. To find the faces of this ROP, we consider the Baxter permutation π^{∗} = ρ(π) obtained by a clockwise rotation of π by 90 degrees. After this rotation, the points ρ(p_{i}) still form a minimal occurrence of 2413. The above arguments imply that there exists in ϕ(π^{∗}) a unique path from ρ(p_{3}) to ρ(p_{1}), which has length 2 and contains only one white vertex w_{2}. Similarly, there exists in ϕ(π^{∗}) a unique path from ρ(p_{4}) to ρ(p_{2}), which has length 2 and contains only one white vertex w_{1}. But Φ(π^{∗}) is the dual orientation O^{∗} (Proposition 4). Let f_{1} and f_{2} be the faces of O that are the duals of w_{1} and w_{2}. We claim that (v_{1}, v_{2}, f_{1}, f_{2}) is a ROP of O.

Consider the superposition of the edges of ϕ(π) and of the edges of ϕ(π^{∗}), rotated by 90 degrees counterclockwise (dashed lines in Fig. 16(c)). Let e_{i} be the edge of O corresponding to the point p_{i}, for 1 ≤ i ≤ 4. By Proposition 4, the dual edge e_{i}^{∗} corresponds to the point ρ(p_{i}). As e_{1}^{∗} starts from w_{2}, the edge e_{1} lies on the left border of the face f_{2} in O. By considering the points p_{2}, p_{3} and p_{4}, one proves similarly that e_{2} lies on the left border of f_{1}, that e_{3} lies on the right border of f_{2} and e_{4} on the right border of f_{1}. In particular, both v_{1} and v_{2} are adjacent to f_{1} and f_{2}.

As p_{4} lies North-East of p_{1}, there is an oriented path from p_{1} to p_{4} in ϕ(π). As p_{1} has outdegree 1, the second vertex on this path is v_{1}, and similarly, its nextto-last vertex is v_{2}. Thus the edge (v_{1}, p_{2}) of ϕ(π) is followed, in clockwise order around v_{1}, by another outgoing edge: this implies that v_{1} is the source of the face f_{1}. At the other end of the path, we observe that the edge (p_{3}, v_{2}) is followed, in clockwise order around v_{2}, by another ingoing edge: thus v_{2} is the sink of f_{2}.

The edge e_{1} ends at v_{1} and is on the left border of f_{2}: as the sink of f_{2} is v_{2} (and v_{2} ≠ v_{1}), v_{1} is a left vertex of f_{2}. Symmetrically, v_{2} is a right vertex of f_{1}.

Hence (v_{1}, v_{2}, f_{1}, f_{2}) is a ROP.

It remains to prove the statement on 3142-avoiding Baxter permutations. Recall that Φ(π^{−1}) = mir(O) and observe that 2413^{−1} = 3142. Note also that the map mir transforms right-oriented pieces on left-oriented pieces. Consequently, a Baxter permutation π avoids 3142 if and only if the orientation Φ(π) has no LOP.

Proof of Proposition 5. The restriction of Φ to 2413-avoiding Baxter permutations is a bijection between these permutations and bipolar plane orientations with no ROP, which transforms standard parameters as described in Theorem 2. By Proposition 15, orientations with no ROP are in bijection with non-separable planar maps (the bijection consists in adding a root-edge from the source to the sink, and forgetting the orientation of all non-root edges). This bijection increases the edge number, the degrees of the source and of the sink by 1, and it transforms the right and left outer degrees of the orientation into the degrees of the faces lying, respectively, to the left and right of the root-edge, minus one. Proposition 5 follows.

7.2. Permutations avoiding 2413 and 3142, and series-parallel maps

We now prove Proposition 6. We have said that a rooted non-separable planar map M is series-parallel if it does not contain K_{4} as a minor. Let M̌ be the corresponding bipolar map. We will say that M̌ itself is series-parallel. By adapting the proof given for graphs in [5], it is not hard to see that a bipolar map M̌ is series-parallel if and only if it can be constructed recursively, starting from the single-edge map, by applying a sequence of series and parallel compositions:

• the series composition of two series-parallel bipolar maps M̌_{1} and M̌_{2} is obtained by identifying the sink of M̌_{1} with the source of M̌_{2} (see Fig. 17 (b)),

• the parallel composition of M̌_{1} and M̌_{2} is obtained by putting M̌_{2} to the right of M̌_{1}, and then identifying the sources of M̌_{1} and M̌_{2}, as well as their sinks (Fig. 17 (c)).

By Proposition 16, a Baxter permutation π avoids both patterns 2413 and 3142 if and only if the corresponding plane bipolar orientation has no ROP nor LOP. We have seen in Proposition 15 that every bipolar map M̌ admits a unique orientation with no ROP and a unique orientation with no LOP, which are respectively the minimal and maximal element of the lattice of orientations of M̌. Thus M̌ admits an orientation with no ROP nor LOP if and only if it has a unique bipolar orientation. Hence, to prove Proposition 6, it suffices to establish the following lemma.

(a)  (b)  (c) t s  s  s t  t M̌_{2} M̌_{1} M̌_{2}  M̌_{1}

Figure 17. The operations that build all series-parallel bipolar maps: (a) taking a single edge, (b) a series composition, (c) a parallel composition.

Lemma 17. A bipolar map M̌ admits a unique bipolar orientation if and only if it is series-parallel.

Proof. From the recursive construction of series-parallel bipolar maps shown in Fig. 17, it is easily checked that such a map admits a unique bipolar orientation (see also [12, Remark 6.2]).

Conversely, assume that M̌ is not series-parallel. This means that the corresponding rooted map M contains K_{4} as a minor. Observe that K_{4} admits exactly two bipolar orientations, show in Fig. 18. From the Extension Lemma of [12], the two bipolar orientations of K_{4} can be extended to two distinct bipolar orientations of M, and thus of M̌.

This concludes the proof of Proposition 6.

Figure 18. The two bipolar orientations of K_{4}. The first one contains a ROP, the other one a LOP.

7.3. A link with a construction of Dulucq, Gire and West

We have seen that the bijection Φ sends 3142-avoiding Baxter permutations onto orientations with no LOP. Clearly, 3142-avoiding Baxter permutations form a subtree of the generating tree T_{b} of Fig. 8: if π avoids 3142, deleting its largest entry will not create an occurrence of this pattern. As Φ is the canonical bijection between T_{b} and the tree T_{o} of orientations, this implies that orientations with no LOP form a subtree of T_{o}. In this subtree, replace every orientation by the underlying bipolar map M̌, and then by the corresponding non-separable rooted map M (this means adding a root-edge to M̌). This gives a generating tree T_{m} for rooted non-separable maps.

In this section we give a description of this tree directly in terms of maps (rather than orientations). We then observe that, up to simple symmetries, this is the tree that was introduced in [14] to prove that non-separable planar maps are in bijection with permutations avoiding the patterns 2413 and 41¯352—equivalently, with 2413avoiding Baxter permutations. In accordance with [14], where the children of a node of the generating tree are not explicitly ordered, we will only describe the non-embedded generating tree. That is, we only explain how to determine the parent of a given rooted non-separable map.

Let M be a rooted non-separable map. Recall that, by convention, the infinite face lies to the right of the root-edge. Let e be the edge following the root-edge in counterclockwise order around the infinite face. We say that M is valid if M \ e is separable.

Proposition 18. Let M be a map of the tree T_{m} having at least three edges, and let e be defined as above. The parent of M in T_{m} is obtained by contracting e if M is valid, and deleting it otherwise.

Proof. Given the description of the parent of an orientation O in the tree T_{o}, and the bijection between orientations with no LOP and rooted non-separable maps, what we have to prove is the following statement: if O is a bipolar orientation with no LOP, e the last edge on its left border, v the starting point of e, and M the associated rooted map, then v has outdegree 1 in O if and only if M is valid.

Assume first that v has outdegree 2 or more. The parent \overline{O} of O in T_{o} is obtained by deleting e from O. The corresponding rooted map (obtained by adding to \overline{O} a root-edge from s to t) is M \ e. It is non-separable (as it admits a bipolar orientation). This means that M is not valid.

Assume now that v has outdegree 1. Let f_{1} be the face lying to the right of e in O. If f_{1} is the infinite face of O, then M \e is separable (the source is a separating vertex), so that M is valid. Now assume that f_{1} is a bounded face, and let s_{1} be its source. We need the following lemma, the proof of which is delayed for the moment.

Lemma 19. The vertex s_{1} is on the left outer border of O.

This lemma implies that s_{1} is a separating vertex in M \ e. Indeed, deleting e and s_{1} from M disconnects the vertex v from the source (Fig. 19). Hence M is valid.

ss_{1} v ef_{1} t

Figure 19. If v has out degree 1, the rooted map M is valid.

Up to minor changes (declaring e as the root edge, and taking the mirror of maps), we have recovered the generating tree T_{m} of rooted non-separable planar maps given in [14, Section 2.1]. In that paper, it is shown that T_{m} is isomorphic to the generating tree of 2413-avoiding Baxter permutations. The canonical (and recursive) bijection between these permutations and non-separable planar maps that results from the existence of this isomorphism is not described explicitly in [14]. Our paper gives a non-recursive description of this bijection (up to elementary symmetries), as a special case of a more general correspondence. Note that the structure of the tree T_{m} is not as simple as that of the more general tree T_{o}. In particular, the description given in [14] involves an unbounded number of labels, while T_{o} is isomorphic to a simple tree T with two integers labels.

We still have to prove Lemma 19. Proof of Lemma 19. Assume the lemma is wrong, and that O is a minimal counterexample (in terms of the edge number). By assumption, s_{1} is not on the left outer border of O. Thus the left face of s_{1}, called f_{2}, is finite, with source s_{2} and sink t_{2} (Fig. 20(a)). The left border of f_{1} is an oriented path P_{0} going from s_{1} to t, which ends with the edge e. Let P_{1} be the leftmost oriented path from s_{1} to t: at each vertex, it takes the leftmost outgoing edge. By planarity, the last edge of P_{1} is also e. The first steps of P_{1} follow the right border of f_{2}, from s_{1} to t_{2}. Let O_{1} be the set of edges lying between P_{1} \ e and P_{0} \ e (both paths are included in O_{1}). Then O_{1} is a bipolar orientation of source s_{1} and sink v.

Observe that t_{2} is not incident to f_{1}, otherwise (s_{1}, t_{2}, f_{1}, f_{2}) would form a LOP. Hence the right face of t_{2}, called f_{3}, is a face of O_{1} (see Fig. 20(b), where only the orientation O_{1} and the faces f_{1} and f_{2} are shown). Let s_{3} be its source, and t_{3} its sink. As s_{1} is the source of O_{1}, there exists an oriented path P from s_{1} to s_{3}. Consider the following two paths that go from s_{2} to t_{3}: the first one follows the left border of f_{2} and then the portion of the left border of f_{3} from t_{2} to t_{3}; the second follows the right border of f_{2} up to s_{1}, then the path P, and finally the right border of f_{3}. Let O_{2} be the plane bipolar orientation formed of the edges lying between these two paths (Fig. 20(c)).

The orientation O_{2} has fewer edges than O, and we claim that it is a counterexample to the lemma. The last edge e′ of its left outer border is also the last edge of the left border of f_{3}. Hence its starting point lies between t_{2} and t_{3}, and has outdegree 1 in O_{2}. The face to the right of e′ is f_{3}. Moreover, the source s_{3} of f_{3} is not on the left outer border of O_{2}, as it is separated from this path by the face f_{2}. Hence O_{2} is a smaller counterexample than O, which yields a contradiction.

8. Final comments

It is natural to study the restriction of our bijection Φ (or its inverse Φ^{−1}) to interesting subclasses of permutations (or orientations), for instance those that are counted by simple numbers. This is the case for alternating and doubly-alternating Baxter permutations [11, 15, 25], or for orientations of triangulations [32]. Recently, we have also discovered a new and intriguing result, which deals with fixed-point-free Baxter involutions. It is easy to construct for them a generating tree, analogous to the tree T_{b} of Baxter permutations (Fig. 8). One goes from a node to its parent by deleting the cycle containing the largest entry. Again, the

f_{2}  f_{1} t_{2}s_{1}  t v s_{2}  t_{3} f_{3} s_{3} P f_{2}  f_{1} s_{1}  t v s_{2} s  e O_{1} t_{2}  t_{3} f_{3} s_{3} f_{2} t_{2}s_{1} s_{2} (a)  (b)  (c)

Figure 20. (a) A plane bipolar orientation with no LOP such that v has outdegree 1. All edges are North-East oriented. The white areas are faces. The source of f_{1} has to be on the left outer border, otherwise, as shown in (b)-(c), a smaller counterexample can be produced.

tree is isomorphic to a generating tree with two labels (like (2)), that encode the number of lr-maxima and rl-maxima. Using the techniques of [7], we have proved that the number of such involutions of length 2n is

\frac{3. 2^{n−1}}{(n + 1)(n + 2)}\binom{2n}{n},

which is also known to count Eulerian planar maps with n edges.

After a 90 degrees rotation, Baxter involutions having no fixed point are described by permutation diagrams that are symmetric with respect to the reflection in the second diagonal and have no point on this diagonal. Via the bijection Φ, they correspond to plane orientations having a single source, but possibly several sinks, lying in the outer face (Fig. 21). It would be interesting to have a combinatorial understanding of the above formula (and of the various refinements of it that we have obtained).

Figure 21. A fixed-point-free Baxter involution, rotated by 90 degrees, and the corresponding plane orientation.

Acknowledgements. We are grateful to Andrei Asinowski, who drew our attention to the connections between Baxter permutations and floorplans.

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N. Bonichon: LaBRI, Université Bordeaux 1, 351 cours de la Libération, 33405 Talence, France

  E-mail address: nicolas.bonichon@labri.fr

M. Bousquet-Mélou: CNRS, LaBRI, Université Bordeaux 1, 351 cours de la Libération, 33405 Talence, France

  E-mail address: mireille.bousquet@labri.fr

É. Fusy: Dept. Mathematics, Simon Fraser University, 8888 University Drive, Burnaby, BC, V5A 1S6, Canada

  E-mail address: eric.fusy@inria.fr